1. Field of the Invention
The present invention relates to a straightness measuring apparatus for a rectilinearly moving body. A rectilinearly moving body such as a pulse stage, a rectilinear motion stage and a rectilinear motion stage is used for feeding a machine tool and for various shape measuring apparatus and the like. Precision of works and measurements is largely dependent upon the degree of linearity of the rectilinearly moving body. A method and apparatus for measuring linearity at a high precision is therefore essential for precision inspection and calibration.
2. Description of the Related Art
A rectilinear motion stage is presently used in various fields such as mechanical industries. A measuring apparatus for inspecting or calibrating a rectilinear motion precision at all times is essential. A method of measuring a motion error of a rectilinear stage is already stipulated in JIS, and a measuring apparatus is also already sold. Generally, errors of rectilinear motion and rotation angle are a translation position error and a rotation angle error. The former parallel two-surface mirrors being mounted on the rectilinearly moving body;
a light source for generating a laser beam;
semi-transparent mirrors for splitting the laser beam into two laser beams;
optical path changing reflection mirrors for changing a direction of each laser beam; and
position sensors for detecting positions of the laser beam reflected from said horizontal, parallel two-surface mirror and said vertical, parallel two-surface mirror,
wherein a direction of said horizontal, parallel two-surface mirror is set so that directions of each normal to a bottom surface of each reflection surface take respectively:
[Expression 3]
mH1=(xe2x88x921/(2), 0, 1/(2)), mH2=(1/(2), 0, 1/(2)) or 
mH1=(1/(2), 0, xe2x88x921/(2)) mxe2x80x2H2(xe2x88x921/(2), 0, 1/(2)) 
a direction of said vertical parallel two-surface mirror is set so that directions of each normal to a bottom surface of each reflection surface take respectively:
[Expression 4]
mV1=(0, 1/(2), xe2x88x921/(2)), mV2=(0, xe2x88x921/(2), 1/(2)) or 
mxe2x80x2V1=(0, 1/(2), 1/(2)), mxe2x80x2V2=(0, xe2x88x921/(2), xe2x88x921/(2)) 
and said light source, said semi-transparent mirrors, said optical path changing reflection mirrors, said corner cube reflection mirrors and said position sensors has three types of error components, including one type of error components of position determination in a rectilinear direction and two types of error components in the plane vertical to the rectilinear direction (horizontal and vertical directions). The latter has three types of error components including pitching, yawing and rolling.
With a general method of measuring a rotation angle error of a rectilinear moving body, two error components of pitching and yawing are measured at the same time with an auto-collimator, and rolling is measured with an electronic level. Apparatus using this method is also sold. However, since the principle of an auto-collimator is utilized, although two types of pitching and yawing error components can be measured at the same time, techniques of measuring three types of rotation errors are not still developed.
The present invention proposes a novel method and apparatus for measuring a translation position error of a rectilinear moving body such as a rectilinear table, in addition to rotation angle errors of pitching, yawing and rolling with one method, by utilizing linearity of a laser beam.
The invention has been made in consideration of the above-described circumstances and aims to provide an apparatus for measuring three components of a rotation angle error including pitching, yawing and rolling of a rectilinear moving body such as a rectilinear table in a non-contact manner and at the same time, by utilizing linearity of a laser beam.
In order to achieve the above object, a straightness measuring apparatus for a rectilinearly moving body of this invention comprises: a horizontal, parallel two-surface mirror positioned in a horizontal plane and including two reflection mirrors whose reflection surfaces face each other and are spaced by a distance (2)d; a vertical, parallel two-surface mirror positioned in a vertical plane and including two reflection mirrors whose reflection surfaces face each other and are spaced by a distance (2)d; a light source for generating a laser beam; semi-transparent mirrors for splitting the laser beam into two laser beams; optical path changing reflection mirrors for changing a direction of each laser beam; and position sensors for detecting positions of the laser beam reflected from the horizontal, parallel two-surface mirror and the vertical, parallel two-surface mirror, wherein a direction of the horizontal, parallel two-surface mirror is set so that directions of each normal to a bottom surface of each reflection surface take respectively:
[Expression 5]
mH1=(xe2x88x921/(2), 0, 1/(2)), mH2=(1/(2), 0, 1/(2)) or 
mH1(1/(2), 0, xe2x88x921/(2)), mxe2x80x2H2=(xe2x88x921/(2), 0, 1/(2))
a direction of the vertical parallel two-surface mirror is set so that directions of each normal to a bottom surface of each reflection surface take respectively:
[Expression 6]
mmV1=(0, 1/(2), xe2x88x921/(2)), mV2=(0, xe2x88x921/(2), 1/(2)) or 
mxe2x80x2V1(0, xe2x88x921/(2), 1/(2)), mxe2x80x2V2(0, xe2x88x921/(2), xe2x88x921/(2))
and the light source, the semi-transparent mirrors, the optical path changing reflection mirrors and the position sensors are fixed to positions outside of the rectilinearly moving body.
An embodiment of the invention will be described in detail with reference to the accompanying drawings.
Referring to FIGS. 1 and 2, reference numeral 1 represents a straightness measuring apparatus 1. The straightness measuring apparatus 1 has a rectilinear moving body 2 whose straightness is to be measured according to the invention. The rectilinear moving body 2 is, for example, a pulse stage or a rectilinear motion table.
In a rectangular coordinate system, a propagation direction of the rectilinearly moving body 2 is a z-axis, a horizontal direction is an x-axis and a vertical direction is a y-axis. Two parallel two-surface mirrors each having reflection surfaces facing each other, i.e., a horizontal, parallel two-surface mirror 3 and a vertical, parallel two-surface mirror, are mounted on the rectilinearly moving body 2. The horizontal, parallel two-surface mirror 3 is constituted of two reflection mirrors MH1 and MH2 whose reflection surfaces are faced each other and spaced by a distance (2)d. The vertical, parallel two-surface mirror 4 is constituted of two reflection mirrors MV1 and MV2 whose reflection surfaces are faced each other and spaced by a distance (2)d. Positioned outside of the rectilinearly moving body 2 are: a light source 5 for generating a laser beam; semi-transparent mirrors BS1 and BS2 for splitting a laser beam from the light source 5 into two beams; optical path changing reflection mirrors M1, M2, M3 and M4 for changing the direction of each beam; two pairs of position sensors PS2 and PS3 for detecting horizontal and vertical positions of each beam reflected from the parallel two-surface mirrors 3 and 4; and position sensors PS1 for detecting the position of a reflection beam from corner cube mirrors CCM. A personal computer (not shown) is also installed for controllably driving the rectilinearly moving body 2 and analyzing position sensor outputs.
It is assumed that while the rectilinearly moving body 2 moves straight, rotation errors of a pitching angle "xgr", yawing angle xcex7 and a rolling angle xcex6 are generated. The rotation errors are obtained from the following expression 7, by representing position errors in the horizontal and vertical directions observed by two position sensors PS2 and PS3 by xH, yH and xV and yV:
[Expression 7]
"xgr"=(xVxe2x88x92yVxe2x88x92xHxe2x88x92yH)/4dxe2x80x83xe2x80x83(1) 
xcex7=(xV+yV+xHxe2x88x92yH)/4dxe2x80x83xe2x80x83(2) 
xcex6=(xVxe2x88x92yVxe2x88x92xH+yH)/4dxe2x80x83xe2x80x83(3)
where it is assumed that the rotation error angles "xgr", xcex7 and xcex6 are sufficiently small.
Next, specific calculation expressions for obtaining the rotation error angles "xgr", xcex7 and xcex6 translation error distances xcex94x and xcex94y will be given.
1. Principle
[0] Description of Light Beam on Mirror Surface
1-1 Reflection of Light Beam at Mirror Surface
A direction of an incidence light beam is expressed by:
[Expression 8]                              l          j                =                  (                                                                      α                  j                                                                                                      β                  j                                                                                                      γ                  j                                                              )                                    (1-1)            
and a direction of a reflection light beam is expressed by:
[Expression 9]                              l                      j            +            1                          =                  (                                                                      α                                      j                    +                    1                                                                                                                        β                                      j                    +                    1                                                                                                                        γ                                      j                    +                    1                                                                                )                                    (1-2)            
where xcex1, xcex2 and xcex3 represent direction cosine, a subscript j represents a reflection order expressed by a natural number, and [ ] represent a matrix or vector.
If an incidence light beam becomes incident upon a point Q (xx, yg, zg) this is expressed by using k as a parameter by:
[Expression 10]
Lj: x=xcex1k+xg, y=xcex2k+yg, z=xcex3k+zgxe2x80x83xe2x80x83(1-3)
By representing a direction of a mirror as:
[Expression 11]                              m          j                =                  (                                                                      λ                  j                                                                                                      μ                  j                                                                                                      ν                  j                                                              )                                    (1-4)            
and by representing a distance to the mirror surface from an origin by h, an expression of the mirror surface is given by:
[Expression 12]
Mj: xcex, x+xcexc, y+xcexd, z=hxe2x80x83xe2x80x83(1-5)
where xcex, xcexc and xcexd represent direction cosine.
Reflection at the mirror is given by a matrix:
[Expression 13]
and a relation between incidence and reflection light beams can be expressed by (refer to FIG. 3):
[Expression 14]
lj+1=M, l,xe2x80x83xe2x80x83(1-7)
where Mj is herein called a reflection matrix.
1-2 Beam Position by Horizontal, Parallel Two-Surface Mirror before Rotation
The layout of the parallel, two-surface mirror and the like and the coordinate system are set as shown in FIG. 4.
FIG. 4 illustrates reflection of a laser beam at the horizontal, parallel two-surface mirror (MH1, 0: first reflection mirror surface, MH2, 0: second reflection mirror surface, LH0: incidence light beam, LH1, 0: reflected light beam by first mirror, LH2, 0: reflected light beam by second mirror, QH0: incidence point on first mirror surface). A direction of an incidence light beam is:
[Expression 15]
lH0=(0, 0, xe2x88x921)xe2x80x83xe2x80x83(2-1)
If the incidence light becomes incident upon a mirror surface point:
[Expression 16]
QH0(xe2x88x92d, 0, 0)xe2x80x83xe2x80x83(2-2)
this is expressed by:
[Expression 17]
LH0: x=xe2x88x92d, y=0, z=xe2x88x92kxe2x80x83xe2x80x83(2-3)
From the expressions (1-1), (1-6) and (1-7), the direction of a light beam reflected by the first mirror is given by:
[Expression 18]                              l                      H1            ,            0                          =                  (                                                    1                                                                    0                                                                    0                                              )                                    (2-4)            
Since the direction of the light beam reflected by the first mirror is lH1, 0 and passes the point QH0, from the expressions (2-4) and (2-2) the following expressions are obtained:
[Expression 19]
LH1, 0: x=k, y=0, z=0xe2x80x83xe2x80x83(2-5)
The direction of a light beam reflected by the second mirror is:
[Expression 20]
and the directions of the two-surface mirror before rotation are:
[Expression 21]                              l                      H2            ,            0                          =                  (                                                    0                                                                    0                                                                                      -                  1                                                              )                                    (2-6)            
Since the two-surface mirror has two mirrors which are parallel and face each other, the expressions (2-7) and (2-8) have inverted signs.
Since the directions of the two-surface mirror are given by the expressions (2-7) and (2-8) and the distance from the origin to the mirror surface is both (2)d, it stands therefore from the expression (1-5):
[Expression 22]
MH1, 0: xe2x88x92xxe2x88x92z=dxe2x80x83xe2x80x83(2-9) 
MH2, 0: x+z=dxe2x80x83xe2x80x83(2-10)
The reflection point RH2, 0 on the second mirror surface is given from the expressions (2-5) and (2-10) by:
[Expression 23]
RH2, 0(d, 0, 0)xe2x80x83xe2x80x83(2-11)
Therefore, the light beam reflected by the second mirror is:
[Expression 24]
LH2, 0: x=d, y=0, z=xe2x88x92kxe2x80x83xe2x80x83(2-12)
It is assumed that the position of a position sensor is:
[Expression 25]
PS H(d, 0, xe2x88x92zn)xe2x80x83xe2x80x83(2-13)
and that ZH is sufficiently larger than the stroke of the table.
2-3 Beam Position by Pitching of Horizontal, Parallel Two-Surface Mirror
A rotation matrix rotating right about the x-axis by "xgr" is given by:
[Expression 26]                              R          ξ                =                  (                                                    1                                            0                                            0                                                                    0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                    0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                              )                                    (3-1)            
By substituting the expressions (6-7) and (6-8) into the expression (3-1), the direction of the rotated mirror is given by:
[Expression 27]                              m                      H1            ,            ξ                          =                                            R              ξ                        ⁢                          m                              H1                ,                0                                              =                      (                                                                                                      -                      1                                        /                                          (                                              2                                            )                                                                                                                                        sin                    ⁢                                          xe2x80x83                                        ⁢                                          ξ                      /                                              (                                                  2                                                )                                                                                                                                                                                    -                      cos                                        ⁢                                          xe2x80x83                                        ⁢                                          ξ                      /                                              (                                                  2                                                )                                                                                                                  )                                              (3-2)                                          m                      H2            ,            ξ                          =                                            R              ξ                        ⁢                          m                              H2                ,                0                                              =                      (                                                                                1                    -                                          /                                              (                                                  2                                                )                                                                                                                                                                                    -                      sin                                        ⁢                                          xe2x80x83                                        ⁢                                          ξ                      /                                              (                                                  2                                                )                                                                                                                                                              cos                    ⁢                                          xe2x80x83                                        ⁢                                          ξ                      /                                              (                                                  2                                                )                                                                                                                  )                                              (3-3)            
Therefore, by substituting the expressions (3-2) and (3-3) into the expression (1-5), the surfaces of the two-surface mirror are given by:
[Expression 28]
MH1, "xgr": xe2x88x92x+sin "xgr" yxe2x88x92cos "xgr" z=dxe2x80x83xe2x80x83(3-4) 
MH2, "xgr": xxe2x88x92sin "xgr" y+cos "xgr" z=dxe2x80x83xe2x80x83(3-5)
From the expressions (6-3) and (3-4), the reflection point by the first mirror is given by:
[Expression 29]
RH1, "xgr" (xe2x88x92d, 0, 0)xe2x80x83xe2x80x83(3-6)
By substituting the expressions (3-2) and (3-3) into the expression (1-6), the reflection matrixes of the rotated two-surface mirror are:
[Expression 30]                              M                      1            ,            ξ                          =                  (                                                    0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  cos                    2                                    ⁢                  ξ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξcos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                                          -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξcos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  sin                    2                                    ⁢                  ξ                                                              )                                    (3-7)                                          M                      2            ,            ξ                          =                  (                                                    0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  cos                    2                                    ⁢                  ξ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξcos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                                          -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξcos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  sin                    2                                    ⁢                  ξ                                                              )                                    (3-8)            
Therefore, from the expressions (3-7) and (6-1), the direction of the light beam reflected by the first mirror is:
[Expression 31]                              l                      H1            ,            ξ                          =                                            M                              1                ,                ξ                                      ⁢                          l              H0                                =                      (                                                                                cos                    ⁢                                          xe2x80x83                                        ⁢                    ξ                                                                                                                                          -                      sin                                        ⁢                                          xe2x80x83                                        ⁢                    ξcos                    ⁢                                          xe2x80x83                                        ⁢                    ξ                                                                                                                                          -                                              sin                        2                                                              ⁢                    ξ                                                                        )                                              (3-9)            
From the expressions (3-8) and (3-9), the direction of the light beam reflected by the second mirror is:
[Expression 32]                              l                      H2            ,            ξ                          =                                            M                              2                ,                ξ                                      ⁢                          l                              H1                ,                ξ                                              =                      (                                                            0                                                                              0                                                                                                  -                    1                                                                        )                                              (3-10)            
Since the direction of the light beam reflected by the first mirror is lH1, "xgr" and passes the reflection point RH1, "xgr", from the expressions (3-9) and (3-6) the following expressions are obtained:
[Expression 33]
LH1, "xgr": x=cos "xgr" kxe2x88x92d, y=xe2x88x92sin "xgr" cos "xgr" k, z=xe2x88x92sin2 "xgr" kxe2x80x83xe2x80x83(3-11)
Since the reflection point on the second mirror is a cross point between MH2, "xgr" and LH1, "xgr", from the expressions (3-5) and (3-11), it stands:
[Expression 34]
RH2, "xgr" (d, xe2x88x922 sin "xgr" d, xe2x88x922 sin2 "xgr" d/cos "xgr")xe2x80x83xe2x80x83(3-12)
2-4 Beam Position by Yawing of Horizontal, Parallel Two-Surface Mirror
A rotation matrix rotating right about the y-axis by xcex7 is given by:
[Expression 35]                              R          η                =                  (                                                                      cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                            0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                    0                                            1                                            0                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  η                                                            0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                              )                                    (4-1)            
By substituting the expressions (6-7) and (6-8) into the expression (7-1), the direction of the rotated mirror is given by:
[Expression 36]
Therefore, by substituting the expressions (4-2) and (4-3) into the expression (1-5), the two-surface mirror is given by:
[Expression 37]
MH1, xcex7: xe2x88x92(cos xcex7+sin xcex7) x+(sin xcex7xe2x88x92cos xcex7) z=dxe2x80x83xe2x80x83(4-4) 
MH2, xcex7: (cos xcex7+sin xcex7) x+(cos xcex7xe2x88x92sin xcex7) z=dxe2x80x83xe2x80x83(4-5)
Since the reflection point by the first mirror is a cross point between MH1, 0 and lH1, 0, from the expressions (6-5) and (4-4), it stands:
[Expression 38]
RH1, xcex7=(xe2x88x92d, 0, (cos xcex7+sin xcex7xe2x88x921)/(cos xcex7xe2x88x92sin xcex7)d)xe2x80x83xe2x80x83(4-6)
By substituting the expressions (4-2) and (4-3) into the expression (1-6), the reflection matrixes of the rotated two-surface mirror are:
[Expression 39]                              M                      1            ,            η                          =                  (                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                            0                                                                                  -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                                                    0                                            1                                            0                                                                                                          -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                            0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                              )                                    (4-7)                                          M                      2            ,            η                          =                  (                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                            0                                                                                  -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                                                    0                                            1                                            0                                                                                                          -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                            0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  η                                                              )                                    (4-8)            
Therefore, from the expressions (7-7) and (6-4), the direction of the light beam reflected by the first mirror is:
[Expression 40]                              l                      H1            ,            η                          =                                            M                              H1                ,                η                                      ⁢                          l              0                                =                      (                                                                                cos                    ⁢                                          xe2x80x83                                        ⁢                    2                    ⁢                    η                                                                                                0                                                                                                                        -                      sin                                        ⁢                                          xe2x80x83                                        ⁢                    2                    ⁢                    η                                                                        )                                              (4-9)            
From the expressions (3-8) and (3-9), the direction of the light beam reflected by the second mirror is:
[Expression 41]                              l                      H2            ,            η                          =                                            M                              H2                ,                η                                      ⁢                          l                              H1                ,                η                                              =                      (                                                            0                                                                              0                                                                                                  -                    1                                                                        )                                              (4-10)            
Since the direction of the light beam reflected by the first mirror is lH1 and passes the reflection point RH1, xcex7, from the expressions (4-9) and (4-6) the following expressions are obtained:
[Expression 42]
LH1, xcex7: x=cos 2 xcex7kxe2x88x92d, y=0, z=xe2x88x92sin 2 xcex7k+(cos xcex7+sin xcex7xe2x88x921)/(cos xcex7xe2x88x92sin xcex7)dxe2x80x83xe2x80x83(4-11)
Since the reflection point on the second mirror is a cross point between MH2xcex7 and LH1, xcex7, from the expressions (4-5) and (4-11), it stands:
[Expression 43]
RH2, xcex7 ({2 (cos xcex7+sin xcex7)xe2x88x921}d, 0, (xe2x88x922 sin 2xcex7+cos xcex7+sin xcex7xe2x88x921) d/(cos xcex7xe2x88x92sin xcex7))xe2x80x83xe2x80x83(4-12)
2-5 Beam Position by Rolling of Horizontal, Parallel Two-Surface Mirror
A rotation matrix rotating right about the z-axis by xcex6 is given by:
[Expression 44]                              R          ζ                =                  (                                                                      cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                                        -                      sin                                        ⁢                                          xe2x80x83                                        ⁢                    ζ                                    ⁢                                      xe2x80x83                                                                              0                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                            0                                                                    0                                            0                                            1                                              )                                    (5-1)            
By substituting the expressions (6-7) and (6-8) into the expression (5-1), the direction of the rotated mirror is given by:
[Expression 45]
mH1, xcex6=Rxcex6mH1, 0=[xe2x88x92cos xcex6/(2), xe2x88x92sin xcex6/(2), xe2x88x921/(2)]xe2x80x83xe2x80x83(5-2) 
mH2, xcex6=Rxcex6mH2, 0=[cos xcex6/(2), sin s xcex6/(2), 1/(2)]xe2x80x83xe2x80x83(5-3)
Therefore, by substituting the expressions (5-2) and (5-3) into the expression (1-5), the surfaces of the two-surface mirror are given by:
[Expression 46]
MH1 xcex6: xe2x88x92cos xcex6xxe2x88x92sin xcex6 yxe2x88x92z=dxe2x80x83xe2x80x83(5-4) 
MH2, xcex6: cos xcex6 x+sin xcex6 y+z=dxe2x80x83xe2x80x83(5-5)
Since the reflection point by the first mirror is a cross point between MH1, xcex6 and LH1, 0, from the expressions (6-5) and (5-4), it stands:
[Expression 47]
RH1, xcex6 (xe2x88x92d, 0, (cos xcex6xe2x88x921) d)xe2x80x83xe2x80x83(5-6)
By substituting the expressions (5-2) and (5-3) into the expression (1-6), the reflection matrixes of the rotated two-surface mirror are:
[Expression 48]                              M                      H1            ,            ζ                          =                  (                                                                                          sin                    2                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  cos                    2                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                                          -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                            0                                              )                                    (5-7)                                          M                      H2            ,            ζ                          =                  (                                                                                          sin                    2                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  cos                    2                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                                          -                    cos                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                            0                                              )                                    (5-8)            
Therefore, from the expressions (5-7) and (6-4), the direction of the light beam reflected by the first mirror is:
[Expression 49]                              l                      H1            ,            ζ                          =                                            M                              H1                ,                ζ                                      ⁢                          l              0                                =                      (                                                                                cos                    ⁢                                          xe2x80x83                                        ⁢                    ζ                                                                                                                    sin                    ⁢                                          xe2x80x83                                        ⁢                    ζ                                                                                                0                                                      )                                              (5-9)            
From the expressions (5-8) and (5-9), the direction of the light beam reflected by the second mirror is:
[Expression 50]                              l                      H2            ,            ζ                          =                                            M                              H2                ,                ζ                                      ⁢                          l                              H1                ,                ζ                                              =                      (                                                            0                                                                              0                                                                                                  -                    1                                                                        )                                              (5-10)            
Since the direction of the light beam reflected by the first mirror is lH1, xcex6 and passes the reflection point RH1, xcex6, from the expressions (5-9) and (5-6) the following expressions are obtained:
[Expression 51]
LH1 xcex6: x=cos xcex6 kxe2x88x92d, y=xe2x88x92sin xcex6 k, z=(cos xcex6xe2x88x921)dxe2x80x83xe2x80x83(5-11 )
Since the reflection point on the second mirror is a cross point between MH2, xcex6 and LH1, xcex6, from the expressions (5-5) and (5-11), it stands:
[Expression 52]
RH2, xcex6 ((2 cos xcex6xe2x88x921) d, 2 sin xcex6 d, (cos xcex6xe2x88x921) d)xe2x80x83xe2x80x83(5-12)
2-6 Beam Position by Vertical, Parallel Two-Surface Mirror before Rotation
The layout of the parallel, two-surface mirror and the coordinate system are set as shown in FIG. 5.
FIG. 5 illustrates reflection of a laser beam at the vertical, parallel two-surface mirror (MV1, 0: first reflection mirror surface, MV2, 0: second reflection mirror surface, L0: incidence light beam, LV1, 0: reflected light beam by first mirror, LV2, 0 reflected light beam by second mirror, QV0: incidence point on first mirror surface, PSV: position sensor). A direction of an incidence light beam is:
[Expression 53]                              l          V0                =                  (                                                    0                                                                    0                                                                                      -                  1                                                              )                                    (6-1)            
If the incidence light becomes incident upon a mirror surface point:
[Expression 54]
QV0 (0, d, 0)xe2x80x83xe2x80x83(6-2)
this is expressed by:
[Expression 55]
LV0: x=0, y=d, z=xe2x88x92kxe2x80x83xe2x80x83(6-3)
From the expressions (6-1), (1-6) and (1-7), the direction of a light beam reflected by the first mirror is given by:
[Expression 56]                              l                      V1            ,            0                          =                  (                                                    0                                                                                      -                  1                                                                                    0                                              )                                    (6-4)            
Since the direction of the light beam reflected by the first mirror is lV0 and passes the point QV0 from the expressions (6-4) and (6-2) the following expressions are obtained:
[Expression 57]
LV1, 0: x=0, y=xe2x88x92k, z=0xe2x80x83xe2x80x83(6-5)
From the expressions (6-4), (1-6) and (1-7), the direction of a light beam reflected by the second mirror is:
[Expression 58]                              l                      V2            ,            0                          =                  (                                                    0                                                                    0                                                                                      -                  1                                                              )                                    (6-6)            
The directions of the two-surface mirror before rotation are:
[Expression 59]                              m                      V1            ,            0                          =                  (                                                    0                                                                                      1                  /                                      (                                          2                                        )                                                                                                                                            -                    1                                    /                                      (                                          2                                        )                                                                                )                                    (6-7)                                          m                      V2            ,            0                          =                  (                                                    0                                                                                                          -                    1                                    /                                      (                                          2                                        )                                                                                                                        1                  /                                      (                                          2                                        )                                                                                )                                    (6-8)            
Since the two-surface mirror has two mirrors which are parallel and face each other, the expressions (6-7) and (6-8) have inverted signs.
Since the directions of the two-surface mirror are given by the expressions (6-7) and (6-8) and the distance from the origin to the mirror surface is both (2)d, it stands therefore from the expression (1-5):
[Expression 60]
MV1, 0: yxe2x88x92z=dxe2x80x83xe2x80x83(6-9) 
MV2, 0: xe2x88x92y+z=dxe2x80x83xe2x80x83(6-10)
The reflection point on the second mirror surface is given from the expressions (6-5) and (6-10) by:
[Expression 61]
RV2, 0 (0, xe2x88x92d, 0)xe2x80x83xe2x80x83(6-11)
Therefore, the light beam reflected by the second mirror is:
[Expression 62]
LV2, 0: x=0, y=xe2x88x92d, z=xe2x88x92kxe2x80x83xe2x80x83(6-12)
It is assumed that the position of a position sensor is:
[Expression 63]
P SV=(0, xe2x88x92d, xe2x88x92zV)xe2x80x83xe2x80x83(6-13)
and that ZV is sufficiently larger than the stroke of the table.
2-7 Beam Position by Pitching of Vertical, Parallel Two-Surface Mirror
A rotation matrix rotating right about the x-axis by "xgr" is given by:
[Expression 64]                              R          ξ                =                  (                                                    1                                            0                                            0                                                                    0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                                    0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                                              )                                    (7-1)            
By substituting the expressions (1-7) and (1-8) into the expression (8-1), the direction of the rotated mirror is given by:
[Expression 65]                              m                      V1            ,            ξ                          =                                            R              ξ                        ⁢                          m                              V1                ,                0                                              =                      (                                                            0                                                                                                                        (                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                +                                                  sin                          ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                                    )                                        /                                          (                                              2                                            )                                                                                                                                                              (                                                                                                    -                            cos                                                    ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                +                                                  sin                          ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                                    )                                        /                                          (                                              2                                            )                                                                                            )                                              (7-2)                                          m                      V2            ,            ξ                          =                                            R              ξ                        ⁢                          m                              V2                ,                0                                              =                      (                                                            0                                                                                                                        (                                                                                                    -                            cos                                                    ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                -                                                  sin                          ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                                    )                                        /                                          (                                              2                                            )                                                                                                                                                              (                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                -                                                  sin                          ⁢                                                      xe2x80x83                                                    ⁢                          ξ                                                                    )                                        /                                          (                                              2                                            )                                                                                            )                                              (7-3)            
Therefore, by substituting the expressions (7-2) and (7-3) into the expression (1-5), the surfaces of the two-surface mirror are given by:
[Expression 66]
Mv1 "xgr": (cos "xgr"+sin "xgr") y+(sin "xgr"xe2x88x92cos "xgr") z=dxe2x80x83xe2x80x83(7-4) 
Mv2, "xgr": xe2x88x92(cos "xgr"+sin "xgr") yxe2x88x92(sin "xgr"xe2x88x92cos "xgr") z=dxe2x80x83xe2x80x83(7-5)
From the expressions (1-3) and (7-4) the reflection point by the first mirror is given by:
[Expression 67]
Rv1, "xgr" (0, d, (1xe2x88x92cos "xgr"xe2x88x92sin "xgr") d/(sin "xgr"xe2x88x92cos "xgr"))xe2x80x83xe2x80x83(7-6)
By substituting the expressions (7-2) and (7-3) into the expression (1-6), the reflection matrixes of the rotated two-surface mirror are:
[Expression 68]                              M                      V1            ,            ξ                          =                  (                                                    1                                            0                                            0                                                                    0                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                                                    0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                              )                                    (7-7)                                          M                      V2            ,            ξ                          =                  (                                                    1                                            0                                            0                                                                    0                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                                                    0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  ξ                                                              )                                    (7-8)            
Therefore, from the expressions (7-7) and (1-1), the direction of the light beam reflected by the first mirror is:
[Expression 69]                              l                      V1            ,            ξ                          =                                            M                              V1                ,                ξ                                      ⁢                          l              V0                                =                      (                                                            0                                                                                                                        -                      cos                                        ⁢                                          xe2x80x83                                        ⁢                    2                    ⁢                    ξ                                                                                                                                          -                      sin                                        ⁢                                          xe2x80x83                                        ⁢                    2                    ⁢                    ξ                                                                        )                                              (7-9)            
From the expressions (7-8) and (7-9), the direction of the light beam reflected by the second mirror is:
[Expression 70]                              l                      V2            ,            ξ                          =                                            M                              V2                ,                ξ                                      ⁢                          l                              V1                ,                ξ                                              =                      (                                                            0                                                                              0                                                                                                  -                    1                                                                        )                                              (7-10)            
Since the direction of the light beam reflected by the first mirror is lV1, "xgr" and passes the reflection point RV1, "xgr", from the expressions (7-9) and (7-6) the following expressions are obtained:
[Expression 71]
Lv1, "xgr": x=0, y=xe2x88x92cos 2 "xgr" k+d, z=xe2x88x92sin 2 "xgr" k+(1xe2x88x92cos "xgr"xe2x88x92sin "xgr")d/(sin "xgr"xe2x88x92cos "xgr")
Since the reflection point on the second mirror is a cross point between MV2, "xgr" and LV1, "xgr", from the expressions (7-5) and (7-11), it stands:
[Expression 72]
RV2 "xgr" (0, {xe2x88x922 (cos "xgr"+sin "xgr")+1}d, (2 sin 2 "xgr"xe2x88x92cos "xgr"xe2x88x92sin "xgr"+1)d/(sin "xgr"xe2x88x92cos "xgr"))xe2x80x83xe2x80x83(7-12)
2-8 Beam Position by Yawing of Vertical, Parallel Two-surface Mirror
A rotation matrix rotating right about the y-axis by xcex7 is given by:
[Expression 73]                              R          η                =                  (                                                                      cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                            0                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                    0                                            1                                            0                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  η                                                            0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                              )                                    (8-1)            
By substituting the expressions (1-7) and (1-8) into the expression (8-1), the direction of the rotated mirror is given by:
[Expression 74]                              m                      V1            ,            η                          =                                            R              η                        ⁢                          m                              V1                ,                0                                              =                      (                                                                                                      -                      sin                                        ⁢                                          xe2x80x83                                        ⁢                                          η                      /                                              (                                                  2                                                )                                                                                                                                                                                    -                      1                                        /                                          (                                              2                                            )                                                                                                                                                              -                      cos                                        ⁢                                          xe2x80x83                                        ⁢                                          η                      /                                              (                                                  2                                                )                                                                                                                  )                                              (8-2)                                          m                      V2            ,            η                          =                                            R              η                        ⁢                          m                              V2                ,                0                                              =                      (                                                                                sin                    ⁢                                          xe2x80x83                                        ⁢                                          η                      /                                              (                                                  2                                                )                                                                                                                                                                                    -                      1                                        /                                          (                                              2                                            )                                                                                                                                        cos                    ⁢                                          xe2x80x83                                        ⁢                                          η                      /                                              (                                                  2                                                )                                                                                                                  )                                              (8-3)            
Therefore, by substituting the expressions (8-2) and (8-3) into the expression (1-5) the two-surface mirror is given by:
[Expression 75]
MV1 xcex7: xe2x88x92sin xcex7 x+yxe2x88x92cos xcex7 z=dxe2x80x83xe2x80x83(8-4) 
MM2 xcex7: sin xcex7 xxe2x88x92y+cos xcex7 z=dxe2x80x83xe2x80x83(8-5)
Since the reflection point by the first mirror is a cross point between MV1, xcex7 and LV1, 0, from the expressions (1-3) and (8-4), it stands:
[Expression 76]
RV1, xcex7 (0, d, 0)xe2x80x83xe2x80x83(8-6)
By substituting the expressions (8-2) and (8-3) into the expression (1-6), the reflection matrixes of the rotated two-surface mirror are:
[Expression 77]                              M                      1            ,            η                          =                  (                                                                                          cos                    2                                    ⁢                  η                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ηcosη                                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  η                                                            0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ηcosη                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                                  sin                    2                                    ⁢                  η                                                              )                                    (8-7)                                          M                      2            ,            η                          =                  (                                                                                          cos                    2                                    ⁢                  η                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ηcosη                                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  η                                                            0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ηcosη                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  η                                                                                                  sin                    2                                    ⁢                  η                                                              )                                    (8-8)            
Therefore, from the expressions (8-7) and (1-1), the direction of the light beam reflected by the first mirror is:
[Expression 78]                              l                      V1            ,            η                          =                                            M                              V1                ,                η                                      ⁢                          l              V0                                =                      (                                                                                sin                    ⁢                                          xe2x80x83                                        ⁢                    ηcos                    ⁢                                          xe2x80x83                                        ⁢                    η                                                                                                                                          -                      cos                                        ⁢                                          xe2x80x83                                        ⁢                    η                                                                                                                                          -                                              sin                        2                                                              ⁢                    η                                                                        )                                              (8-9)            
From the expressions (8-8) and (8-9), the direction of the light beam reflected by the second mirror is:
[Expression 79]                              l                      V2            ,            η                          =                                            M                              V2                ,                η                                      ⁢                          l                              V1                ,                η                                              =                      (                                                            0                                                                              0                                                                                                  -                    1                                                                        )                                              (8-10)            
Since the direction of the light beam reflected by the first mirror is lV1 xcex7, and passes the reflection point RV1, xcex7, from the expressions (8-9) and (8-6) the following expressions are obtained:
[Expression 80]
LV1 xcex7: x=sin xcex7 cos xcex7 k, y=xe2x88x92cos xcex7 k+d, z=xe2x88x92sin xcex7 kxe2x80x83xe2x80x83(8-11)
Since the reflection point on the second mirror is a cross point between MV2, xcex7 and LV2, xcex7, from the expressions (8-5) and (8-11), it stands:
[Expression 81]
RV2, xcex7 (2 sin xcex7 d, xe2x88x92d, xe2x88x922 sin2 xcex7 d/cos xcex7)xe2x80x83xe2x80x83(8-12)
2-9 Beam Position by Rolling of Vertical, Parallel Two-Surface Mirror
A rotation matrix rotating right about the z-axis by xcex6 is given by:
[Expression 82]                              R          ζ                =                  (                                                                      cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                            0                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                            0                                                                    0                                            0                                            1                                              )                                    (9-1)            
By substituting the expressions (1-7) and (1-8) into the expression (9-1), the direction of the rotated mirror is given by:
[Expression 83]                              m                      V1            ,            ζ                          =                                            R              ζ                        ⁢                          m                              V1                ,                0                                              =                      (                                                                                                      -                      sin                                        ⁢                                          xe2x80x83                                        ⁢                                          ζ                      /                                              (                                                  2                                                )                                                                                                                                                              cos                    ⁢                                          xe2x80x83                                        ⁢                                          ζ                      /                                              (                                                  2                                                )                                                                                                                                                                                    -                      1                                        /                                          (                                              2                                            )                                                                                            )                                              (9-2)                                          m                      V2            ,            ζ                          =                                            R              ζ                        ⁢                          m                              V2                ,                0                                              =                      (                                                                                sin                    ⁢                                          xe2x80x83                                        ⁢                                          ζ                      /                                              (                                                  2                                                )                                                                                                                                                              cos                    ⁢                                          xe2x80x83                                        ⁢                                          ζ                      /                                              (                                                  2                                                )                                                                                                                                                              1                    /                                          (                                              2                                            )                                                                                            )                                              (9-3)            
Therefore, by substituting the expressions (9-2) and (9-3) into the expression (1-5), the surfaces of the two-surface mirror are given by:
[Expression 84]
MV1, xcex6: xe2x88x92sin xcex6 x+cos xcex6 yxe2x88x92z=dxe2x80x83xe2x80x83(9-4) 
MV2, xcex6: sin xcex6 xxe2x88x92cos xcex6 y+z=dxe2x80x83xe2x80x83(9-5)
Since the reflection point by the first mirror is a cross point between MV1, xcex6 and LV1, 0, from the expressions (1-5) and (9-4), it stands:
[Expression 85]
RV1 xcex6 (0, d, (cos xcex6xe2x88x921)d)xe2x80x83xe2x80x83(9-6)
By substituting the expressions (9-2) and (9-3) into the expression (1-6), the reflection matrixes of the rotated two-surface mirror are:
[Expression 86]                              M                      V1            ,            ζ                          =                  (                                                                                          cos                    2                                    ⁢                  ζ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  sin                    2                                    ⁢                  ζ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                            0                                              )                                    (9-7)                                          M                      V2            ,            ζ                          =                  (                                                                                          cos                    2                                    ⁢                  ζ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                      sin                  ⁢                                      xe2x80x83                                    ⁢                  ζcos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                  sin                    2                                    ⁢                  ζ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                                                                          -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  ζ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  ζ                                                            0                                              )                                    (9-8)            
Therefore, from the expressions (9-7) and (1-1), the direction of the light beam reflected by the first mirror is:
[Expression 87]                              l                      V1            ,            ζ                          =                                            M                              v1                ,                ζ                                      ⁢                          l              0                                =                      (                                                                                sin                    ⁢                                          xe2x80x83                                        ⁢                    ζ                                                                                                                                          -                      cos                                        ⁢                                          xe2x80x83                                        ⁢                    ζ                                                                                                0                                                      )                                              (9-9)            
From the expressions (9-8) and (9-9), the direction of the light beam reflected by the second mirror is:
[Expression 88]                              l                      V2            ,            ζ                          =                                            M                              V2                ,                ζ                                      ⁢                          l                              V1                ,                ζ                                              =                      (                                                            0                                                                              0                                                                                                  -                    1                                                                        )                                              (9-10)            
Since the direction of the light beam reflected by the first mirror is lV1, xcex6 and passes the reflection point RV1, xcex6, from the expressions (9-9) and (9-6) the following expressions are obtained:
[Expression 89]
LV1, xcex6: x=sin xcex6 k, y=xe2x88x92cos xcex6 k+d, z=(cos xcex6xe2x88x921)dxe2x80x83xe2x80x83(9-11)
Since the reflection point on the second mirror is a cross point between MV2, xcex6and LV1, xcex6, from the expressions (9-5) and (9-11), it stands:
[Expression 90]
RV2, xcex6 (2 sin xcex6d, (1xe2x88x922 cos xcex6)d, (cos xcex6xe2x88x921)d)xe2x80x83xe2x80x83(9-12)
2-10 Calculation of Rotation Angle Error from Beam Position Detection Amount
The beam positions reflected by the horizontal, parallel two-surface mirror and detected with the position sensor PSH are represented by XH and YH, and the beam positions reflected by the vertical, parallel two-surface mirror and detected with the position sensor PSV are represented by XV and YV,
First, the beam position detection amount by the position sensor is calculated assuming that there is a single rotation error of the parallel two-surface mirror.
When the horizontal, parallel two-surface mirror has a pitch angle of "xgr", the detection amount of the reflected beam position is given from the expressions (2-13) and (3-12) by:
[Expression 91]
XH "xgr"=0xe2x80x83xe2x80x83(10-1 a) 
YH "xgr"=xe2x88x922 sin "xgr"dxe2x80x83xe2x80x83(10-1 b)
When the horizontal, parallel two-surface mirror has a yaw angle xcex7, the detection amount of the reflected beam position is given from the expressions (2-13) and (4-12) by:
[Expression 92]
XH xcex7=2 (cos xcex7+sin xcex7xe2x88x921)dxe2x80x83xe2x80x83(10-2 a) 
YH xcex7=0xe2x80x83xe2x80x83(10-2 b)
When the horizontal, parallel two-surface mirror has a roll angle xcex6, the detection amount of the reflected beam position is given from the expressions (2-13) and (5-12) by:
[Expression 93]
XH xcex6=2 (cos xcex6xe2x88x921)dxe2x80x83xe2x80x83(10-3 a) 
YH xcex6=2 sin xcex6dxe2x80x83xe2x80x83(10-3 b)
When the vertical, parallel two-surface mirror has a pitch angle of "xgr", the detection amount of the reflected beam position is given from the expressions (6-13) and (7-12) by:
[Expression 94]
XV "xgr"=0xe2x80x83xe2x80x83(10-4 a) 
YV "xgr"=xe2x88x922(cos "xgr"+son "xgr"xe2x88x921)dxe2x80x83xe2x80x83(10-4 b)
When the vertical, parallel two-surface mirror has a yaw angle xcex7, the detection amount of the reflected beam position is given from the expressions (6-13) and (8-12) by:
[Expression 95]
XV xcex7=2 sin xcex7dxe2x80x83xe2x80x83(10-5 a) 
YV xcex7=0xe2x80x83xe2x80x83(1 0-5 b)
When the horizontal, parallel two-surface mirror has a roll angle xcex6, the detection amount of the reflected beam position is given from the expressions (6-13) and (9-12) by:
[Expression 96]
XV xcex6=2 sin xcex6dxe2x80x83xe2x80x83(10-6 a) 
YV xcex6=xe2x88x922(cos xcex6xe2x88x921)dxe2x80x83xe2x80x83(10-6 b)
Next, the beam position detection amount by the position sensor is calculated assuming that there is a composite rotation error of the parallel two-surface mirror.
If there are the rotation angle errors of all the pitch angle "xgr", yaw angle xcex7 and roll angle xcex6 at the same time, the X coordinate value of the beam position reflected by the horizontal, parallel two-surface mirror is obtained through addition of the expressions (10-1a), (10-2a) and (10-3a) by:
[Expression 97]
XH T=2(cos xcex7+sin xcex7+cos xcex6xe2x88x922)dxe2x80x83xe2x80x83(10-7)
and similarly the Y coordinated value is obtained through addition of the expressions (10-1b), (10-2b) and (10-3b) by:
[Expression 98]
YH T=2(xe2x88x92sin "xgr"+sin xcex6)dxe2x80x83xe2x80x83(10-8)
If there are the rotation angle errors of all the pitch angle "xgr", yaw angle xcex7 and roll angle xcex6 at the same time, the X coordinate value of the beam position reflected by the vertical, parallel two-surface mirror is obtained through addition of the expressions (10-4a), (10-5a) and (10-6a) by:
[Expression 99]
XV T=2(sin xcex7+sin xcex6)dxe2x80x83xe2x80x83(10-9)
and similarly the Y coordinated value is obtained through addition of the expressions (10-4b), (10-5b) and (10-6b) by:
[Expression 100]
YV T=xe2x88x922(cos "xgr"+sin "xgr"+cos xcex6xe2x88x922)dxe2x80x83xe2x80x83(10-10)
The expressions (10-7) to (10-9) are rewritten as:
[Expression 101]
cos xcex7+sin xcex7+cos xcex6=XH T/(2 d)+2xe2x80x83xe2x80x83(10-11) 
xe2x88x92sin "xgr"+sin xcex6=YH T/(2 d)xe2x80x83xe2x80x83(10-12) 
sin xcex7+sin xcex6=XV T/(2 d)xe2x80x83xe2x80x83(10-13) 
cos "xgr"+sin "xgr"+cos xcex6=xe2x88x92YV T/(2 d)+2xe2x80x83xe2x80x83(10-14)
As cos xcex6 is eliminated through substraction between the expressions (10-13) and (10-14), it becomes:
[Expression 102]
xe2x88x92(sin "xgr"+cos "xgr")+(sin xcex7+cos xcex7)=(XH T+YV T)/(2 d)xe2x80x83xe2x80x83(10-15)
As sin xcex6 is eliminated through substraction of (10-13)-(10-12), it becomes:
[Expression 103]
sin "xgr"+sin xcex7=(XV Txe2x88x92YH T)/(2 d)xe2x80x83xe2x80x83(10-16)
Since the rotation angle error is generally sufficiently small, approximate expressions of sin xcex8≈xcex8 and cos xcex8≈1 for small xcex8 are incorporated. The expressions (10-12) and (10-13) and the expressions (10-15) and (10-16) are therefore expressed as:
[Expression 104]
xe2x88x92"xgr"+xcex6=YH T/(2 d)xe2x80x83xe2x80x83(10-17) 
xcex7+xcex6=XV T/(2 d)xe2x80x83xe2x80x83(10-18) 
xe2x88x92"xgr"+xcex6=(XH T+YV T)/(2 d)xe2x80x83xe2x80x83(10-19) 
"xgr"+xcex7=(XV Txe2x88x92YH T)/(2 d)xe2x80x83xe2x80x83(10-20)
From the expressions (10-19) and (10-20), "xgr" and xcex7 are obtained which are substituted into the expression (10-17) or (10-18) to obtain xcex6:
[Expression 105]
"xgr"=(1/4 d) (xe2x88x92XH Txe2x88x92YH T+XV Txe2x88x92YV T)xe2x80x83xe2x80x83(10-21) 
xcex7=(1/4 d) (XH Txe2x88x92YH T+XV T+YV T)xe2x80x83xe2x80x83(10-22) 
xcex6=(1/4 d) (xe2x88x92XH T+YH T+XV Txe2x88x92YV T)xe2x80x83xe2x80x83(10-23)
Next, by obtaining a position displacement of light reflected from the corner cube mirror on the position sensor PS1, a translation position error of the rectilinearly moving body 2 can be obtained.
(Straightness Measuring Optical System)
In the measuring apparatus shown in FIGS. 1 and 2, an incidence laser beam is split int a transmission laser beam and a reflection laser beam by the beam splitter BS1. The reflection light from BS1 becomes incident upon the corner cube mirror fixed to the rectilinear motion table, reflected by CCM, returns in parallel to the incidence light, and is detected with the position sensor PS1. The reflection light in CCM is always reflected at the position symmetrical to the apex of CCM. This position is detected with PS1 at a twofold sensitivity when the rectilinear motion table has a translation error in the vertical and horizontal directions. Namely, the following relation is satisfied:
[Expression 106]
DX=2DX 
DV=2DYxe2x80x83xe2x80x83(1-1)
where DX and DY represent values detected with PS1 and are horizontal and vertical translation errors of the rectilinear motion table. In this case, even if the table has a rotation angle error, the measured values of translation errors are not influenced because of the characteristics of the corner cube mirror.
The laser beam transmitted through BS1 is further split into a reflection beam and a transmission beam. The reflection beam from BS2 is sequentially reflected by MH1 and MH2 constituting the horizontal, parallel two-surface mirror, and reaches the position sensor PS2. The transmission beam from SB2 is reflected by M3 and M4 to be made in parallel to the optical axis (z-axis), thereafter sequentially reflected by MV1 and MV2 constituting the vertical, parallel two-surface mirror, and reaches the position sensor PS3. The rotation angle error components of pitching, yawing and rolling are obtained from the detection outputs of PS2 and PS3 according to the expressions (10-21) to (10-23), as described earlier.
With the method described above, it is possible to obtain the rotation angle errors of pitching, yawing and rolling of a rectilinear motion table from the positions of beams reflected from two vertical and horizontal, parallel two-surface mirrors disposed at an incidence angle of 45xc2x0. It is also possible to obtain a straightness including two translation error components of the rectilinear motion table, by incorporating a conventional method of measuring horizontal and vertical translation errors by utilizing the corner cube mirror.
In the optical system shown in FIGS. 1 and 2, a higher sensitivity is realized if a multi-reflection system is adopted by fixing a roof prism or rectangular two-surface mirror outside of the rectilinear motion table and sandwiching the corner cube mirror and parallel two-surface mirrors. A higher sensitivity is also realized if a sensor such as a four-division photosensor is used as the position sensor. In the optical system shown in FIGS. 1 and 2, PS1 is disposed at a position opposite to PS2 and PS3. If this layout is inconvenient for a measuring work or the measuring apparatus becomes not compact, a roof prism or rectangular two-surface mirror may be fixed on one side to dispose all position sensors on the same side.
FIGS. 6 and 7 illustrate a relation between mirrors and a light beam according to another embodiment of the invention. A direction of the surface of a first reflection mirror Mxe2x80x2H2, 0 constituting a horizontal, parallel two-surface mirror shown in FIG. 6 is given by:
[Expression 107]      m          H2      ,      0        xe2x80x2    =      (                                                      -              1                        /                          (                              2                            )                                                            0                                                  1            /                          (                              2                            )                                            )  
whereas a direction of the a second reflection mirror Mxe2x80x2H1, 0 is given by:
[Expression 108]      m          H1      ,      0        xe2x80x2    =      (                                        1            /                          (                              2                            )                                                            0                                                                -              1                        /                          (                              2                            )                                            )  
A direction of the surface of a first reflection mirror Mxe2x80x2H2, 0 constituting a vertical, parallel two-surface mirror shown in FIG. 7 is given by:
[Expression 109]      m          V2      ,      0        xe2x80x2    =      (                            0                                                  1            /                          (                              2                            )                                                                        1            /                          (                              2                            )                                            )  
whereas a direction of the a second reflection mirror Mxe2x80x2V1, 0 is given by:
[Expression 110]      m          v1      ,      0        xe2x80x2    =      (                            0                                                                -              1                        /                          (                              2                            )                                                                                      -              1                        /                          (                              2                            )                                            )  
[Effects of the Invention]
According to the present invention, it is possible to measure the rotation angle errors including pitching, yawing and rolling components of a rectilinear motion table at the same time with a single method by using two parallel two-surface mirrors perpendicular relative to a laser beam.
FIG. 1 An illustrative diagram showing the plan structure of a measuring apparatus for measuring rectilinear motion and rotation angle errors of a rectilinearly moving body according to the invention.
FIG. 2 An illustrative diagram showing the front structure of the measuring apparatus for measuring rectilinear motion and rotation angle errors of a rectilinearly moving body according to the invention.
FIG. 3 An illustrative diagram showing a direction of a reflection mirror surface.
FIG. 4 An illustrative diagram showing a principle of measuring a rotation angle error.
FIG. 5 An illustrative diagram showing a principle of measuring a rotation angle error.
FIG. 6 An illustrative diagram showing a principle of measuring a rotation angle error.
FIG. 7 An illustrative diagram showing a principle of measuring a rotation angle error.